Here's a problem that occurs in automatic program analysis. for a set of variables x1; : : : ; xn, you are given some equality constraints, of the form xi = xj and some disequality constraints,

of the form xi 6= xj . is it possible to satisfy all of them?

for instance, the constraints

x1 = x2; x2 = x3; x3 = x4; x1 6= x4

cannot be satis ed. give an ef cient algorithm that takes as input mconstraints over n variables

and decides whether the constraints can be satisfied.

1- is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis. 2- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis. 3- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in part b?

What additional information could be used to prove abc =mqr using sas? check all that apply.

Which equation shows y= 3 4 x− 5 2 in standard form? a 4x−3y=10 b 3x−4y=10 c 3x−4y=−10 d 4x−3y=−10